A R K I V F O R M A T E M A T I K B a n d 4 n r 43
C o m m u n i c a t e d 6 D e c e m b e r 1961 b y O. FI~OSTMAlq
A property o f bounded normal operators in Hilbert space
B y GORAN BJORCK a n d YIDAR THOMI~E
Let H be a complex Hilbert space (i.e. a complete inner product space over the complex numbers) with scalar product ( . , . ), norm I1" II, and identity ope- rator I. For any bounded linear operator B in H we denote the spectrum b y sp (B) and the point-spectrum by psp (B). The smallest circular disc containing sp (B) is denoted by dB, the boundary of dB b y cB, and its center and radius b y zB and RB, respectively. For a plane set S we denote the convex hull by cony (S).
For u and v in H we define
sin (u, v) = (1 - ](u, v)]2 ~ 89
\
Ilull~jlvll~!(If u or v is zero, we define sin (u,v)=O.)
The purpose of this note is to prove and discuss the following Theorem 1. Let N be a bounded normal operator in H. Then
sup/11Null2 I(Nu'-~)12~ = R ~ ,
=+o ~ Ilu[I 2 Ilnll ~ ! (1)
or equivalently, SUPu.0 ~ sin (u, Nu) = RN. (2)
Pro@ We may evidently suppose t h a t RN> 0. For any complex ~t and a n y u E H with I l u l l = l ,
IlNull 2 - I ( N u , u)I s : II N ~ - ~ull 2 - I ( U u ,
u) - ~ 12,(3)
Hence, taking 2 = zm
sup/!lNull~ I<~u-'~u)l~X = sup < N u - z N u 2 - 1 ( N u , ~)-~NI:)
=+o \ l l u l l ~ Ilull 4 ! u =1
< sup
I I N u - z N u l l ~ = l l N - z N I l l 2 = R ~ ,r
Nu]]=l
since the norm and the spectral radius of the normal operator N - zN I are equal.
We will now exhibit a sequence
unEH
such t h a tand
Ilu.l[+l, ]
II Nu. - u. II +R , when
( N u . , u.) ~ zN
n-+ ~ . (5)
The theorem will then be proved.
We first estabhsh t h a t zNE conv (sp (N)N cN). Otherwise, we could find a clo- sed half-circle
h~CN
having no point in common with sp (N). Since h and sp (N) are compact, they would have a positive distance, and it would follow t h a t sp (N) is contained in a smaller circular disc t h a n dN.Thus there exist two or three distinct complex numbers ~t 1 .. . . . ~trE sp (N) such t h a t ( r = 2 or 3)
12i - ZNr 12 = R2N' r }
ZN=~m~2~
with ~ m ~ = l and m l > 0 .i = l i=1
(6)
r <0~+ ( i = l , r) i n H w i t h B y the properties of spectrum there exist sequences
.tun fn:l
. . . ,liu )l I = l, (7)
IINu2>-: o1,,= +11+0 when
n - - > ~ . (8) Since N is normal and the 2~ are distinct, (8) implies(u~ >,u~))-->0 for i#?" when n - - > ~ . (9) Finally we define u, = Z [ : I m~ 89 u~ ). The verification of (5) is immediate, using (6),
(7), (8) and (9). This completes the proof.
The proof can, of course, also be carried through b y means of the spectral re- presentation of N (for notation, cf. [2]),
N = f z E ( d x d y ) , z
= x + iy.
I n this case the problem reduces to finding sup
I(da),
wherea n d ~/~ is the set of positive measures on sp (N) of the form
da=(E(dxdy)u,u)
with
]lull
= 1. We m a y also write2I(da) =
f f l z - ~ ] 2 da(z) da($).
One of the authors ([1], Theorems 5 and 6) has studied the problem of maxi-
ARKIV FOR MATEMATIK. B d 4 n r 4 3
mizing this integral for all positive measures on sp (N) with total mass 1, using essentially the above argument.
I t follows from the proof t h a t the sup in (1) and (2) is attained if and only if we can find u E H , I l u l l = l , such t h a t
[[ N u - ZN U 112 = f [ Z -- ZNI 2 (E(dxdy) u, u) = R~N,
(Nu, u) = f z(E(dxdy) u, u) = za.
These equations express t h a t ( E ( d x d y ) u , u ) has its s u p p o r t on cN and its center of g r a v i t y in za. I n particular, the sup is attained if the points ~t in (6) and (8) can be chosen as eigenvalues (and the sequences u~ ) as constant sequences of ei- genvectors), i.e. if ZN E cony (psi) (N) n ca). This situation occurs for instance if sp ( N ) = psp (N), which is, of course, always the case if H is finite-dimensional.
If N is self-adjoint, so t h a t sp (N) is a subset of the real line, then RN is half the diameter of the spectrum, and the sup in (1) and (2) can be attained only in the case discussed above.
I t is easy to construct an example of a (necessarily non self-adjoint) normal operator N in a (necessarily infinite-dimensional) Hilbert space H such t h a t psp (N) is e m p t y and still the sup in (1) and (2) is attained. (One can, for example, arrange t h a t sp (/V)=ca.)
We will a p p l y Theorem 1 and its proof to the following problem: Given u , / E H, w h a t can be said a b o u t the spectrum of a n y bounded normal operator with N u = ] ? The following result follows at once from (4).
Corollary 1. Let u , / e H , Ilull = 1, and let r = II/11 sin (/,u). Let N be a bounded nor- mal operator such that N u = / . Then
o 2
r" ~< R a - I ( / , u) - ZN[ 2. (10) I n particular RN >~ r.
Corollary 1 has the following consequence for self-adjoint operators:
Corollary 2. Let u , / e H , I[ul] = l, r=[[/] I sin (/,u), and let (/,u) be real. Let A be a sel]-ad]oint operator such that A u = ]. Then conv(sp(A)) contains some closed interval [(/, u ) - :~, (/, u ) + flJ w i t h ~ = r ~
Corollary 2 is immediately obtained from Corollary 1. I n fact, conv(sp(A))
= [zA- RA, z~ +RA] and we have to prove t h a t
(ZA + RA -- (/, U)) ((/, U) -- ZA § RA) >t r 2, which in this case is equivalent to (10).
As a trivial consequence of Corollary 2 we get:
Corollary 3. Let u , / e H, HuH = 1, r =
II/Jl
sin (/, u), and let (/, u) > O. Let A be a non- negative, sel/-ad]oint operator with A u = / . Thens u p 2 / > (1, u) + r2/(l, u ) =
II 1113/(/,
u).). G s p ( A )
This estimate, which can also be easily o b t a i n e d directly, is stronger t h a n t h e obvious ones, (/,u) a n d ]]/]]. I n fact, we h a v e b y the C a u c h y inequality,
(/,u)
<II/11
<II/llV(/,u).
W e conclude this p a p e r b y discussing t h e following problem: G i v e n u , / E H with I] u]l = 1, can we always find a n o r m a l o p e r a t o r N w i t h N u = / s u c h t h a t we get e q u a l i t y in (10)? T h e answer is contained in the following
Theorem 2. Let u, I E H ,
Ilull
= 1 , and let r =II111
sin ( l , u ) > 0 . Assume that the com- plex number z and the positive number R satislyr3 = R2 - I(1, u) - zt 3. ( 1 1 )
Then there exists a normal operator N with N u = / such that RN = R and ZN= Z. I n particular, /or given u , / E H, I] u II = 1, it is always possible t o / l u g a normal operator N such that N u = / and RN = r. I n this case, necessarily zN = z = (/, u).
Proo/. W e first p r o v e the t h e o r e m for the case z = 0 a n d (/,u) real. Consider t h e subspace G in H , s p a n n e d b y u a n d /. W e will p r o v e t h a t we can find t w o orthogonal vectors Y~I a n d 92 such t h a t
I n fact, (12) is solved b y
u=~01+~2' 1 (12)
/ = Ry~ 1 - RyJ2. j
Using the definition of r, we get f r o m (11)
T h e linear o p e r a t o r A defined b y
A ( X l ~ l + x 2 ~ 3 § ) = x l R ~ l - x 2 R y ) 3 , v_L G,
t h e n satisfies the conditions in T h e o r e m 2. T h e o p e r a t o r A is even self-adjoint.
T h e general case can now be reduced to the case considered above. P u t
~/'
= e ' ~ zu), where the real n u m b e r 0 is chosen such t h a tARKIV FOR MATEMATIK. B d 4 n r 4 3
(/',
u ) = ego((/, u) - z)is real. W e e a s i l y f i n d (cf. (3)) t h a t
r ' = II/']1 sin ( u , l ' ) = Ilill sin ( u , / ) = r,
a n d t h e r e f o r e r '2 = R 2 - (/', u) 2.
W e can t h u s f i n d a s e l f - a d j o i n t o p e r a t o r A w i t h c o n v ( s p ( A ) ) e q u a l t o t h e closed i n t e r v a l [ - R, R] a n d such t h a t A u = ~'. T h e n o r m a l o p e r a t o r N = e -~ ~ A + z I t h e n satisfies t h e c o n d i t i o n s in T h e o r e m 2.
T h e a u t h o r s a r e i n d e b t e d t o P r o f e s s o r L a r s H S r m a n d e r for s t i m u l a t i n g discus- sions on t h e c o n t e n t s of t h i s p a p e r .
University o/Stockholm, Sweden.
R E F E R E N C E S
1. BJORCK, G., Distributions of positive mass, which maximize a certain generalized energy integral. Ark. Mat. 3, 255-269 (1955).
2. RIESZ, F., and Sz.-NAGY, B., Legons d'analyse fonctionelle. Budapest, 1952.
Tryckt den 24 juli 1962
Uppsala 1962. Almqvist & Wiksells Boktryckeri AB
